3.577 \(\int \frac{\sqrt{x}}{\sqrt [3]{x}+x} \, dx\)

Optimal. Leaf size=108 \[ 2 \sqrt{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

[Out]

2*Sqrt[x] + (3*ArcTan[1 - Sqrt[2]*x^(1/6)])/Sqrt[2] - (3*ArcTan[1 + Sqrt[2]*x^(1/6)])/Sqrt[2] - (3*Log[1 - Sqr
t[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2]) + (3*Log[1 + Sqrt[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2])

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Rubi [A]  time = 0.0827125, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {1584, 341, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ 2 \sqrt{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(x^(1/3) + x),x]

[Out]

2*Sqrt[x] + (3*ArcTan[1 - Sqrt[2]*x^(1/6)])/Sqrt[2] - (3*ArcTan[1 + Sqrt[2]*x^(1/6)])/Sqrt[2] - (3*Log[1 - Sqr
t[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2]) + (3*Log[1 + Sqrt[2]*x^(1/6) + x^(1/3)])/(2*Sqrt[2])

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\sqrt [3]{x}+x} \, dx &=\int \frac{\sqrt [6]{x}}{1+x^{2/3}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^{5/2}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt{x}-3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt{x}-6 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt{x}+3 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )-3 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt{x}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt{2}}\\ &=2 \sqrt{x}-\frac{3 \log \left (1-\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}\\ &=2 \sqrt{x}+\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \tan ^{-1}\left (1+\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \log \left (1-\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.006702, size = 24, normalized size = 0.22 \[ -2 \sqrt{x} \left (\, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-x^{2/3}\right )-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(x^(1/3) + x),x]

[Out]

-2*Sqrt[x]*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -x^(2/3)])

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Maple [A]  time = 0.005, size = 71, normalized size = 0.7 \begin{align*} 2\,\sqrt{x}-{\frac{3\,\sqrt{2}}{2}\arctan \left ( 1+\sqrt [6]{x}\sqrt{2} \right ) }-{\frac{3\,\sqrt{2}}{2}\arctan \left ( -1+\sqrt [6]{x}\sqrt{2} \right ) }-{\frac{3\,\sqrt{2}}{4}\ln \left ({ \left ( 1+\sqrt [3]{x}-\sqrt [6]{x}\sqrt{2} \right ) \left ( 1+\sqrt [3]{x}+\sqrt [6]{x}\sqrt{2} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(x^(1/3)+x),x)

[Out]

2*x^(1/2)-3/2*arctan(1+x^(1/6)*2^(1/2))*2^(1/2)-3/2*arctan(-1+x^(1/6)*2^(1/2))*2^(1/2)-3/4*2^(1/2)*ln((1+x^(1/
3)-x^(1/6)*2^(1/2))/(1+x^(1/3)+x^(1/6)*2^(1/2)))

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Maxima [A]  time = 1.70134, size = 112, normalized size = 1.04 \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) - \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(x^(1/3)+x),x, algorithm="maxima")

[Out]

-3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) - 3/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*x^(1/6))
) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x^(1/3) + 1) - 3/4*sqrt(2)*log(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 2*sqrt(
x)

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Fricas [A]  time = 1.27114, size = 404, normalized size = 3.74 \begin{align*} 3 \, \sqrt{2} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1} - \sqrt{2} x^{\frac{1}{6}} - 1\right ) + 3 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4} - \sqrt{2} x^{\frac{1}{6}} + 1\right ) + \frac{3}{4} \, \sqrt{2} \log \left (4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4\right ) + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(x^(1/3)+x),x, algorithm="fricas")

[Out]

3*sqrt(2)*arctan(sqrt(2)*sqrt(sqrt(2)*x^(1/6) + x^(1/3) + 1) - sqrt(2)*x^(1/6) - 1) + 3*sqrt(2)*arctan(1/2*sqr
t(2)*sqrt(-4*sqrt(2)*x^(1/6) + 4*x^(1/3) + 4) - sqrt(2)*x^(1/6) + 1) + 3/4*sqrt(2)*log(4*sqrt(2)*x^(1/6) + 4*x
^(1/3) + 4) - 3/4*sqrt(2)*log(-4*sqrt(2)*x^(1/6) + 4*x^(1/3) + 4) + 2*sqrt(x)

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Sympy [A]  time = 1.93226, size = 110, normalized size = 1.02 \begin{align*} 2 \sqrt{x} - \frac{3 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt [6]{x} + 4 \sqrt [3]{x} + 4 \right )}}{4} + \frac{3 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt [6]{x} + 4 \sqrt [3]{x} + 4 \right )}}{4} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} - 1 \right )}}{2} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(x**(1/3)+x),x)

[Out]

2*sqrt(x) - 3*sqrt(2)*log(-4*sqrt(2)*x**(1/6) + 4*x**(1/3) + 4)/4 + 3*sqrt(2)*log(4*sqrt(2)*x**(1/6) + 4*x**(1
/3) + 4)/4 - 3*sqrt(2)*atan(sqrt(2)*x**(1/6) - 1)/2 - 3*sqrt(2)*atan(sqrt(2)*x**(1/6) + 1)/2

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Giac [A]  time = 1.10828, size = 112, normalized size = 1.04 \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) - \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(x^(1/3)+x),x, algorithm="giac")

[Out]

-3/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*x^(1/6))) - 3/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*x^(1/6))
) + 3/4*sqrt(2)*log(sqrt(2)*x^(1/6) + x^(1/3) + 1) - 3/4*sqrt(2)*log(-sqrt(2)*x^(1/6) + x^(1/3) + 1) + 2*sqrt(
x)